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Chapter 12: Q6RE (page 844)

\(a)\) What does it mean for a set of operators to be functionally complete\(?\)

\(b)\)Is the set \(\{ {\bf{ + }}, \cdot \} \) functionally complete\(?\)

\(c)\)Are there sets of a single operator that are functionally complete\(?\)

Short Answer

Expert verified

\(a)\)All Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\).

\(b)\)\(\{ {\bf{ + }}, \cdot \} \)is not functionally complete.

\(c)\)The set of operations is called functionally complete.

Step by step solution

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01

Using the Boolean expression

(a)

If all Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\), then the set \(A\) of operations is said to be functionally complete.

Therefore, \(\{ {\bf{ + }}, \cdot \} \) is functionally complete.

02

Check whether it is functionally complete or not

(b)

If all Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\), then the set \(A\) of operations is said to be functionally complete.

Therefore, \(\{ {\bf{ + }}, \cdot \} \) is not functionally complete.

03

Check whether it is functionally complete or not

(c)

If all Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\), then the set \(A\) of operations is said to be functionally complete.

Therefore, \(\{ \uparrow \} \) is a single operator which is functionally complete.

So, a single operator can be functionally complete.

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Most popular questions from this chapter

Design a circuit that implements majority voting for five individuals.

Find the sum-of-products expansions of the Boolean function \({\bf{F}}\left( {{\bf{x, y, z}}} \right)\) that equals 1 if and only if

a) \({\bf{x = 0}}\)

b) \({\bf{xy = 0}}\)

c) \({\bf{x + y = 0}}\)

d) \({\bf{xyz = 0}}\)

Construct a multiplexer using AND gates, OR gates, andinverters that has as input the four bits\({{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}\)and the two control bits\({{\bf{c}}_{\bf{o}}}\)and\({{\bf{c}}_{\bf{1}}}\). Set up the circuit so that\({{\bf{x}}_{\bf{i}}}\)is the output, where iis the value of the two-bit integer\({{\bf{(}}{{\bf{c}}_{\bf{1}}}{{\bf{c}}_{\bf{o}}}{\bf{)}}_{\bf{2}}}\).The depthof a combinatorial circuit can be defined by specifyingthat the depth of the initial input is 0 and if a gate has ndifferent inputs at depths\({{\bf{d}}_{\bf{1}}}{\bf{,}}{{\bf{d}}_{\bf{2}}}{\bf{,}}.....{\bf{,}}{{\bf{d}}_{\bf{n}}}\),respectively, then its outputs have depth equal to max\({\bf{(}}{{\bf{d}}_{\bf{1}}}{\bf{,}}{{\bf{d}}_{\bf{2}}}{\bf{,}}.....{\bf{,}}{{\bf{d}}_{\bf{n}}}{\bf{) + 1}}\); this value is also defined to be the depth of the gate. The depth of a combinatorial circuit is the maximum depth of the gates in the circuit.

Use a \({\bf{3 - }}\)cube \({{\bf{Q}}_{\bf{3}}}\) to represent each of the Boolean functions in Exercise \(5\) by displaying a black circle at each vertex that corresponds to a \({\bf{3 - }}\)tuple where this function has the value \({\bf{1}}\).

Is it always true that \((x \odot y) \odot z{\bf{ = }}x \odot (y \odot z)\)\(?\)
See all solutions

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